Well...all drives have onboard controllers, be they HDD, SATA, or NVMe, but NVMe works differently compared to the other two. Compact Flash is the only one which comes to mind which integrates everything, which is how you can take a CF card, use an 80 pin IDE cable, and use it as a native IDE device in a computer from the 90s. HDD controllers are pretty dumb, by design:
You still need controller "drivers" in the OS as the OS needs to actually see the drive, but with the much more simplified nature of NVMe compared to SATA means that no external "controller" as such is required so no specialized drivers are required. If the drive manufacturer has their own drivers, which most do, they should be used as they can be more efficient since the manufacturers can tailor them for each drive, but as far as a measurable performance difference ...Maybe if you had one of the new 64 core EPYC processors then yes, but in a desktop, no.
Compared to SATA